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------------------------------------------------ Add the polynomials [tex]$7x^3 + 2x^2 - 5x - 7$[/tex] and [tex]$-5x^2 + x^3 + 4x - 5$[/tex].

Sure! Let's add the two polynomials step by step.

We are given two polynomials:
1. [tex]$ 7x^3 + 2x^2 - 5x - 7 $[/tex]
2. [tex]$ x^3 - 5x^2 + 4x - 5 $[/tex]

To add these polynomials, we will combine like terms. Like terms are terms that have the same variable raised to the same power.

1. Combine the [tex]$ x^3 $[/tex] terms:
- [tex]$ 7x^3 $[/tex] from the first polynomial
- [tex]$ x^3 $[/tex] from the second polynomial

Adding these, [tex]$ 7x^3 + x^3 = 8x^3 $[/tex]

2. Combine the [tex]$ x^2 $[/tex] terms:
- [tex]$ 2x^2 $[/tex] from the first polynomial
- [tex]$-5x^2$[/tex] from the second polynomial

Adding these together, [tex]$ 2x^2 - 5x^2 = -3x^2 $[/tex]

3. Combine the [tex]$ x $[/tex] terms:
- [tex]$-5x$[/tex] from the first polynomial
- [tex]$4x$[/tex] from the second polynomial

Adding these, [tex]$-5x + 4x = -x$[/tex]

4. Combine the constant terms:
- [tex]$-7$[/tex] from the first polynomial
- [tex]$-5$[/tex] from the second polynomial

Adding these, [tex]$-7 - 5 = -12$[/tex]

Now, put all the terms together:

The sum of the two polynomials is [tex]\[ 8x^3 - 3x^2 - x - 12 \][/tex]

That's the final simplified result!